Topic 17: Vectors; dot productTopic 17 NotesJeremy Orloff17 Topic 17: Vectors; dot productOld, compressed version of topic 17 notes.VectorsTwo views: First the geometric and then the analytic.Geometric viewVector = length and direction: (Discuss scaling, scalars)?? ??(same vector)Length: denoted |A|, also called magnitude or normAddition: (head to tail) Subtraction: either tail to tail or A+(−B)??A//B77A + B??A//−B//BTTA − BTTAnalytic or algebraic viewPlace the tail of A at the origin ⇒ the coordinates of the head determine A:A = ha1, a2i = a1i + a2j.(a1, a2)??A//a1iOOa2j??−→P Q•PQYou’ve seen the vectors i and j in physics. Theyhave coordinates i = h1, 0i, j = h0, 1i//iOOjNotation and terminology1, (a1, a2) indicate as point in the plane.2. ha1, a2i indicates the vector from the origin to the point (a1, a2). Of course, thisvector can be translated anywhere and ha1, a2i = a1i + a2j.3. For A = a1i + a2j, a1and a2are called the i and j components of A. Note: theyare scalars.5.−→P =−−→OP is the vector from the origin to P .6. In print we will often drop the arrow and just use the bold face to indicate a vector, i.e. P ≡−→P.7. Scalars: a real number is a scalar, you can use it to scale a vector.Length: |A| =pa21+ a22Addition: (a1i + a2j) + (b1i + b2j) = (a1+ b1)i + (a2+ b2)j117 TOPIC 17: VECTORS; DOT PRODUCT 2⇔ ha1, a2i + hb1, b2i = ha1+ b1, a2+ b2i−−→PQ =−→Q −−→P (i.e.−−→PQ is the displacement from P to Q) –understand this geomet-rically and analalyticallyDot product (scalar product)Geometric definition: A · B = |A||B|cos θ??A//Bθ//B??ATTA − BθAlgebraic viewA · B = a1b1+ a2b2(Hard to get geometrically)proof: Law of cosines: (won’t do in class)|A − B|2= |A|2+ |B|2− 2|A||B|cos θ⇒ (a21+ a22) + (b21+ b22) − ((a1− b1)2+ (a2− b2)2) = 2|A||B|cos θ⇒ a1b1+ a2b2= |A||B|cos θ. QEDAlgebraic law: A · (B + C) = A · B + A · C.Follows from the algebraic view of dot product.Example: Find the dot product of A and B.i) |A| = 2, |B| = 5, θ = π/4. answer: (draw picture) A · B = |A||B|cos θ =10√2/2 = 5√2..ii) A = i + 2j, B = 3i + 4j. answer: A · B = 1 · 3 + 2 · 4 = 11.Unit vectorsSpecial vectors: i and j. Note i · i = 1 = j · j and i · j = 0.Unit vector: u: |u| = 1. Often indicate bybu.Example: Are the following unit vectors? (i) i + j, (ii)35i +45j. answer: (i) No. (ii) Yes.Example: Find two unit vectors parallel to 2i + 3j.answer: (2i + 3j)/√13, −(2i + 3j)/√13Components or projection:A · u = |A|cos θA · A = |A|2A ⊥ B ⇔ A · B = 0//bu;;A|A|cos θ = component of A in direction ofbuθThe component of A in the direction ofbu is A ·bu. (Note: it is a scalar.)For a non-unit vector: the component of A in the direction of B is the component ofA in the direction ofbu =B|B|.Example: Find the component of A in the direction of B.i) |A| = 2, |B| = 5, θ = π/4. answer: (draw picture)ii) A = i + 2j, B = 3i + 4j. answer: Unit vector in direction of B isB|B|=35i +45j⇒ component is A · B/|B| = 3/5 + 8/5 = 11/5.Trig identity cos(β − α) = cos α cos β − sin α sin β17 TOPIC 17: VECTORS; DOT PRODUCT 3Unit vectors: u = cos α i + sin α j, v = cos β i + sin β jAngle between them is θ = β − αGeometric: u · v = |u||v|cos θ = cos θ = cos(β − α)Analytic: u · v = u1v1+ u2v2= cos α cos β + sin α sin β.77GGαβ − α(cos α, sin α)(cos β, sin β)Example: P = (−5, 0), Q = (1, 3) ⇒−−→PQ = 6i + 3j = h6, 3i.Example: Show−−→PQ +−−→QR +−−→QP = 077__PQRExample: Find 2 unit vectors parallel to v = 3i − 4j.|v| = 5: u1=34i −45j, u2= −u1.Example: Let A = (1, 2), B = (2, 3) and C = (2, −1). Find the cosine of ∠BAC.Let θ be the angle ⇒ cos θ =−−→AB ·−−→AC|AB||AC|.−−→AB = h1, 1i,−−→AC = h1, −3i⇒ cos θ =1 − 3√2√10= −2√20=1√5.<<//ABCθExample: Velocities are vectorsA river flows at 3mph and a rower rows at 6mph. What heading shouldhe use to get straight across a river?Need sin θ =36⇒ θ = π/6Answer: Head at angle of π/6 radians upstream from straight across.__boat//riverOOθSame question with river=2 mph, row=2√2 mph:⇒ sin θ =22√2⇒ θ = π/4.Same question with river=6 mph, row=3 mph:⇒ sin θ =63⇒ No such θExample: Show the sum of the medians of a triangle = 0.Median of AB = P =12(A + B) ⇒ CP =12(B + A) − C.Likewise:−−→BQ =12(A + C) − B,−−→AR =12(B + C) − A.⇒ Sum of medians =−−→CP +−−→BQ +−−→AR = 0.•••ABCPRQThree dimensionsExactly the same except we have a third coordinate:a1i + a2j + a3k = ha1, a2, a3i17 TOPIC 17: VECTORS; DOT PRODUCT 4Example:Show A = (4, 3, 6), B = (−2, 0, 8), C = (1, 5, 0)are the vertices of a right triangle.Two legs of the triangle are−−→AC = h−3, 2, −6iand−−→AB = h−6, −3, 2i⇒−−→AC ·−−→AB = 18 − 6 − 12 = 0 ⇒
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